On the spectrum of differential operators under Riemannian coverings

TL;DR

This paper investigates how the spectrum of differential operators on Riemannian manifolds behaves under coverings, establishing conditions under which spectra are contained or essential, extending known results to broader classes of operators.

Contribution

It provides new spectral comparison results for differential operators under Riemannian coverings, especially for infinite sheeted amenable coverings and their impact on essential spectra.

Findings

Spectrum of the base manifold's operator is contained in the essential spectrum of the lifted operator.

Infinite, amenable coverings imply the lifted operator's spectrum is essential with no finite-multiplicity eigenvalues.

Amenability of the covering relates to spectral preservation of Schrödinger operators on closed manifolds.

Abstract

For a Riemannian covering $p \colon M_{2} \to M_{1}$, we compare the spectrum of an essentially self-adjoint differential operator $D_{1}$ on a bundle $E_{1} \to M_{1}$ with the spectrum of its lift $D_{2}$ on $p^{*}E_{1} \to M_{2}$. We prove that if the covering is infinite sheeted and amenable, then the spectrum of $D_{1}$ is contained in the essential spectrum of any self-adjoint extension of $D_{2}$. We show that if the deck transformations group of the covering is infinite and $D_{2}$ is essentially self-adjoint (or symmetric and bounded from below), then $D_{2}$ (or the Friedrichs extension of $D_{2}$) does not have eigenvalues of finite multiplicity and in particular, its spectrum is essential. Moreover, we prove that if $M_{1}$ is closed, then $p$ is amenable if and only if it preserves the bottom of the spectrum of some/any Schr\"{o}dinger operator, extending a result due to Brooks.